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Average Case Network Lifetime on an Interval with Adjustable Sensing Ranges

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Abstract

Given n sensors on an interval, each of which is equipped with an adjustable sensing radius and a unit battery charge that drains in inverse linear proportion to its radius, what schedule will maximize the lifetime of a network that covers the entire interval? Trivially, any reasonable algorithm is at least a 2-approximation for this Sensor Strip Cover problem, so we focus on developing an efficient algorithm that maximizes the expected network lifetime under a random uniform model of sensor distribution. We demonstrate one such algorithm that achieves an expected network lifetime within 12 % of the theoretical maximum. Most of the algorithms that we consider come from a particular family of RoundRobin coverage, in which sensors take turns covering predefined areas until their battery runs out.

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Notes

  1. We use the terms strip and interval interchangeably to refer to the one-dimensional coverage region.

  2. Please see Remark 1 and the Open Problems section for a discussion of conditions under which the latter assumption can be lifted without loss of generality.

  3. We will occasionally abuse notation by using T to refer to either the lifetime of the system, or the random variable giving the lifetime of a sensor. The precise meaning should be clear from context.

  4. Note that the first and last intervals, U k (0)=[0,2−k−1] and U k (2k)=[1−2−k−1,1], respectively, are only half as wide as the others, all of which have width 2−k.

  5. If the integers from 1 to 2k−1 were placed in a binary tree, η(i) tell us how high up i is in that tree.

  6. We let Γ k (0) be the set of sensors assigned to U k (0) or U k (2k), and have those run RoundRobin on U after all other sensors complete. Their contribution to the network lifetime becomes negligible as k→∞, so we omit it from our calculations.

  7. The inequality is justified by the preceding argument that in practice, the actual load balancing will work at least this well.

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Authors and Affiliations

  1. Department of Computer and Information Science, Brooklyn College of the City University of New York, 2900 Bedford Ave., Brooklyn, New York, NY, 11210, USA

    Amotz Bar-Noy

  2. Department of Mathematics & Statistics, Smith College, Clark Science Center, 44 College Lane, Northampton, MA, 01063, USA

    Ben Baumer

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  1. Amotz Bar-Noy
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  2. Ben Baumer
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Correspondence to Ben Baumer.

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A preliminary version of this work appeared in [2].

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Bar-Noy, A., Baumer, B. Average Case Network Lifetime on an Interval with Adjustable Sensing Ranges. Algorithmica 72, 148–166 (2015). https://doi.org/10.1007/s00453-013-9853-5

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